CS6: Have a Product Code but cannot get a Serial Code anywhere

Multi-volume (Educational) licenes issued a product code that had to be redeemed for an actual serial number with which you could activate the software.   See if this helps you.

https://helpx.adobe.com/download-install/using/student-teacher-serial-product-code.html

Adobe does not ask for money for support (for new products, not for support) so you were most likely talking to a scammer

This is the way to get Adobe support...

Be sure to remain signed in with your Adobe ID before accessing the link below
Do be aware that Adobe's staffing is impacted by the virus, so wait times are long
-you MUST make sure that your browser does NOT block ads, scripts, or pop-ups
-you MUST also allow 'cookies' in your web browser for the link to work
This link will go directly to the chat option
Type "Agent" in the chat box to bypass the chatbot and be connected to a person
-Meaning type in the word Agent without the quote " " marks
https://helpx.adobe.com/contact.html?rghtup=autoOpen
- or by telephone https://helpx.adobe.com/contact/phone.html
.
Adobe does NOT do support by email, so beware of answering anyone who sends you a private message
-click for more information about scammers https://tinyurl.com/10791730

p.s. your acrobat serial number may be different than the suite's serial number.

Adobe wouldn't low-ball you and ask for money to retrieve a serial number.  If you registered the serial when you got the product, you should be able to log into your account and get that serial.  If CS5.5 has a working serial you may be able to get the download here: https://prodesigntools.com/adobe-direct-download-links.html , but understand that whether it is CS5.5 or CS6, you may run into issues on a newer computer that may prevent you from using the program. 

If all you simply need is a way to convert your file, Adobe has an online tool to convert a PDF to Word here: https://www.adobe.com/acrobat/online/pdf-to-word.html . That may be of interest for you.

Support is long finished for CS6, but I'm puzzled. You say you have a serial for "Premiere Design CS6". There wasn't one of them, but perhaps it's "Design & Web Premium". So what is the license you are looking for, that sounds like what you need?